What a corner allows

Unfold any flat model — a paper crane works well — and look at a point where several creases meet. Some creases stand up as ridges (mountains), some sink as valleys.

Count them: around every such point, mountains and valleys differ by exactly two. And if you add every other angle around the point, you get 180°. These two facts (Maekawa’s and Kawasaki’s) decide whether a single meeting point can lie flat.

They are why a road map fights back when you refold it the wrong way, and why a pattern that passes at every point can still jam as a whole: layers from far apart can collide. Deciding that for a whole sheet is, in general, a very hard problem.